Extremal K\"ahler metrics on projectivised vector bundles

We prove the existence of extremal, non-csc, K\"ahler metrics on certain unstable projectivised vector bundles $\P (E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic K\"ahler classes. In particular, the vector bundles $E \to M$ under consideration are assumed to split as a direct sum of stable subbundles $E=E_1 \oplus \dots \oplus E_s$ all having different Mumford-Takemoto-slope, e.g. $\mu(E_1) > \dots > \mu(E_s)$.